Killing the GCH everywhere with a single real

Abstract

Shelah-Woodin investigate the possibility of violating instances of $GCH$ through the addition of a single real. In particular they show that it is possible to obtain a failure of $CH$ by adding a single real to a model of $GCH$, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate $GCH$ at all infinite cardinals by adding a single real to a model of $GCH.$ Our assumption is the existence of an $H(\kappa^{+3})$-strong cardinal, by work of Gitik and Mitchell it is known that more than an $H(\kappa^{++})$-strong cardinal is required.